Tuesday 8-5-96
The relevant section in the book is 23.2
Homework problems : 20, 25, 31, 37, 39, 41, 51, 81
Constructive and destructive interference of light waves is also the reason why thin films of liquid (such as soap bubbles, or oil or gas on a wet road) show colorful patterns. This is known as thin-film interference, because it is the interference of light waves reflecting off the top surface of a film with the waves reflecting from the bottom surface. To obtain a nice colored pattern, the thickness of the film has to be on the order of the wavelength of light.
Consider the case of a thin film of oil floating on water. Thin-film interference can take place if these two light waves interfere constructively: (1) the light from the air reflecting off the top surface, and (2) the light travelling from the air, through the oil, reflecting off the bottom surface, travelling back through the oil and out into the air again. An important consideration in determining whether these waves interfere constructively or destructively is the fact that whenever light reflects off a surface of higher index of refraction, a 180 degree phase shift in the wave is introduced.
A 180 degree phase shift when n2 > n1
On the other hand, when light is travelling in a medium of higher index of refraction than the medium it reflects from (n2 < n1), there is no phase change.
Light in air, reflecting off just about anything (glass, water, oil, etc.) will undergo a 180 degree shift. On the other hand, light in oil, which has a higher n than water does, will have no phase shift if it reflects off an oil-water interface.
To get constructive interference, the two reflected waves have to be shifted by an integer multiple of wavelengths. This must account for any phase shift introduced by a reflection off a higher-n material, as well as for the extra distance travelled by the wave travelling down and back through the film. With the oil film example, constructive interference will occur if the film thickness is 1/4 wavelength, 3/4 wavelength, 5/4, etc. Destructive interference occurs when the thickness is 1/2 wavelength, 1 wavelength, 3/2 wavelength, etc.
In the case of 1/4 wavelength, the wave reflected off the top surface is shifted by 1/2 a wavelength by the reflection. The wave travelling through the film has no phase shift, but travels a total down-and-back distance of 1/2 wavelength, meaning that it will be in phase with the wave reflected from the top. On the other hand, if the film thickness is 1/2 wavelength, the first wave gets a 1/2 wavelength shift and the other gets a wavelength shift...these would cancel each other out.
Note that one has to be very careful in throwing around terms like wavelength, because the wavelength depends on the index of refraction. The film thickness, for constructive interference in the example above, has to be 1/4 (or 3/4 or 5/4 or ...) the wavelength of the light in the oil. This wavelength is related to the wavelength in air by lambda (oil) = lambda (air) / n(oil).
The cancellation (destructive interference) of reflected light waves is utilized to make non-reflective coatings. The coating is usually put over glass, and the coating material generally has an index of refraction less than that of glass. In that case, then, both reflected waves have a 180 degree phase shift, and a film thickness of 1/4 wavelength (in the film) would produce a net shift of 1/2 wavelength, resulting in cancellation.
For non-reflective coatings, then, the minimum film thickness required is
t = lambda (coating) / 4 = lambda (air) / 4 n
where n is the index of refraction of the coating material.